Why are we so political and sectarian that we don't even mention certain sons of Imam Ali just because they have the names Abu Bakr, 'Umar, and 'Uthman, even if they sacrificed their lives for Islam in Karbala?

He mentioned them and we don't ignore them they are in our books. Umar r.a was named after Umar bn Um Salama r.a .. Othman was named after Othman ibn Mathoon r.a and there was no Abu Bakr. His name was Muhammad and his nickname sunnis gave him is abu bakr